I have not read the letter in full and have no idea what it is about but I have read so many volumes of old English theology that the language comes easy to me.
I think 'conciliating their affections' is straightforward. It means pacifying their holy sense. The context is in reproof. While reproving someone's wrong you also satisfy their sense of right, if you acknowledge that the person being corrected has some good things in them. It means while opposing things, they also knew deep inside to be wrong, Darby is described as accepting other things they knew were not wrong. This pacified their heavenly or holy feelings while correcting their evil parts which their own consciences also did.
The next line about making an appeal to their common Head this also seems straightforward. The author seems to be saying that Darby no longer pacified their sense of right but judged them unfairly. Therefore they appeal to Darby to consider Christ which is the common Head of the body of Christ, i.e. the head of the invisible Church who judges all men fairly. Or more simply they appealed to Christ.
The whole section your are pondering seems to be saying Darby was rebuking someone and at first the rebuke was received, but then he went too far and lost his authority as an older brother. Now those who were receptive of his rebuke and leadership were no longer. However as they did not have enough confidence to rebuke him back, as they seem to hold him in some kind of esteem, they appeal to Christ directly hoping that Darby might be smitten in his own conscience and might act like an older brother again.
This seem much like a dispute within a family. Its about a struggle between respect and hurt feelings that need to be pacified if any restoration can be made in the relationship.